Solving One Variable Equations

A variable in mathematics is a symbol that is used in place of a value. The value
of a variable depends on the other values in the expression or equation. As a result,
the variable can at times change accordingly.

Variables can be letters, Greek symbols or combinations of many other symbols.

Expressions

An expression in mathematics refers to a combination of mathematical symbols and
operators and operators. For example,

is an expression since it combines the symbols 5 and 3 (in this case the symbols
are real numbers) and the mathematical addition operator (+).

An expression can also consist of variables as in the example below

where x is a variable because it represents a value which we do no know at
the moment.

Expressions can also be written without operators as long as variables are involved,
for example, the following are also expressions:

where t, x, y and α are variables.

Equations

When you include an equals sign (=) in a mathematical expression, you end up with
an equation. On either side of the equals sign is an expression which leads to the
definition of an equation as a mathematical statement that asserts the equality
of two expressions. In simpler terms, an equation is a statement which tells us
that one thing is equal to another.

Here are some examples of equations:

Solving Equations

Now that we have established that an equation is a statement of equality, we're
able to solve for unknown variables in equations. Solving equations is a fundamental
theorem of Algebra and Mathematics as a whole since all the different aspects incorporate
some sort of solving equations.

Example 1

Given the equation below, solve for the unknown variable:

Step 1

In the above, we currently aren't aware of what value the variable x represents,
and so our task to find out what that is.

The first step is to check how many variables we have and how many known values
we have. In this example we only have one variable x and two known values
3 and 6.

Step 2

Since all the variables are on the left hand side of the equals sign, let's focus
on the expression on that side for a minute,

We're adding some number x to 3. If we look at the right hand side expression,
6, we realize that we need to find some number x to which you add
3 to get 6.

Step 3

From Elementary Algebra, we know that if x + 3 = 6, then we collect like terms and
shift the numbers to one side and leave the variable on the other side. To achieve
this in this particular example, we subtract 3 from both sides of the equation as
below:

therefore,

Example 2

Solve for y in the following equation:

Step 1

This example isn't so different from the previous one. The expression on the left
containing a variable is equal to the expression on the right.

Thus in this example we're finding a number represented by the variable y
which when doubled and then 4 subtracted from it, will equal to 12.

Step 2

The first step in solving the above is to collect like terms, we need to put all
the numbers on one side and leave only the variable on the other side. We need to
exercise caution here since, there is multiplication involved.

By adding 4 to each side, we get rid of it from the left hand side of the
equation

Step 3

So now only 2 is left on the left hand side. We simply can't add or subtract
to move it to the right hand side of the equation since it is multiplied to y.
To get rid of it, we pide through by 2 as below:

and we end up with

Therefore y = 8 is our solution.

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